■ak takPak calculate the next dimension (a is increased by 1). The converged latent vectors are computed for a given a for all k, then a is incremented by 1 and the process is repeated until a = A. The model generated can be used to monitor future batches by storing pak, wak, and dk for a = 1 ,••• k = 1,... ,K.

As data are collected from the new batch and stored as row vectors xjT, the values for rak, tak, and X(a+1)fc are computed at time k for a = 1, ■ • • , A

0.03r

0.03r

Variable No.

Figure 6.67. Variable contributions to T2 when SPE chart signals out-of-control status at the 305th measurement.

Variable No.

Figure 6.67. Variable contributions to T2 when SPE chart signals out-of-control status at the 305th measurement.

by using rafc = X-akPak, tafc = [t„(fc_i) dk rafc]wafc, (6.126)

Xfa+ l)fc = XIfc ~ tafePlfe (6-127) The prediction error is computed as a efc = - £ tafcp£fc (6.128)

The score and error values at each k can be plotted for MSPM of the batch. Since no missing data estimation is required in AHPCA, the control limits are calculated directly using the residuals and scores from the model building stage.

Example. AHPCA-based SPM framework is illustrated using the same simulated data set of fed-batch penicillin production presented in Section 6.4.1. Two main steps of this framework can be expressed as model development stage using a historical reference batch database that defines normal operation and process monitoring stage by making use of the model developed for monitoring a new batch.

AHPCA model development stage: AHPCA model is developed from a data set of equalized/synchronized (Figures 6.42 and 6.43), unfolded and scaled 37 good batches (each containing 14 variables 764 measurements resulting

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